package My::Module;

shift;              # throw away my own package name
our $fh   = shift;
{...}               # check to see it's a good filehandle

sub later {
    print $fh, 'Arbitrary text.';
};

1;
[download]

I think you rather meant:

package My::Module;

our $fh;

sub import {
    shift;              # throw away my own package name
    $fh   = shift;
    #{...}              # check to see it's a good filehandle
}

sub later {
    print $fh 'Arbitrary text.';
}

1;
[download]

As for what arguments to allow, I would not go to the trouble of allowing bareword filehandles (STDOUT) or filehandle names as strings ('STDOUT'), but a file name passed as a plain string (use My::Module '/var/foo.log';) might make sense, IMHO, in which case the module would open the corresponding handle.  (You could use ref \$fh in the module's import routine to tell apart...)

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